Existence of Equilibrium and Comparative Statics
in Differentiated Goods Cournot Oligopolies
Steffen H. Hoernig∗
Universidade Nova de Lisboa, Portugal
February 2003
Abstract
We show that under horizontal differentiation pure symmetric Cournot
equilibria exist if firms react to a rise in competitors’ output in such a way
that their market price does not rise. This condition is related to strategic complementarity, but not to convexity or differentiability. We rule out
multiple equilibria under some additional conditions and discuss stability and
regularity of equilibria with ordinal methods. Following entry equilibrium
price decreases only if competitors’ outputs enter inverse demand aggregated
into a single number, otherwise it may increase even in stable equilibria. The
comparative statics of quantities and profits for all equilibria are given.
JEL classification: C62, D43, L13
Keywords: Cournot oligopoly, horizontal product differentiation, comparative statics, stability, entry, ordinal methods
∗
Faculdade de Economia, Universidade Nova de Lisboa, Travessa Estêvão Pinto, 1099-032 Lisboa, Portugal. tel +351-21-3801645, fax +351-21-3886073, email [email protected].
1
Introduction
Since Cournot’s early contribution his model of oligopoly has received more and more
attention, and nowadays is a basic building block of applied work on a wide range
of topics involving imperfect competition. Its usefulness depends on two features:
First, existence and uniqueness of equilibria at the market stage must be easily
established, and second, comparative statics results should be readily available. In
the context of homogeneous goods both these aspects have been treated extensively,
whereas for differentiated goods there are much fewer results available. The existing
literature on this subject is very rich, and to keep the introduction short we will
discuss it below in a separate section 6.
We impose the condition that firms react to a rise in competitors’ quantities by
adjusting their own production in such a way that their market price does not rise
(condition A). Doing so, output may increase or decrease, but must not decrease
too strongly. This condition is formulated without making use of differentiability
or convexity assumptions, rather is formulated in ordinal terms as a single-crossing
condition.1 If goods are substitutes, and under standard regularity conditions such
as continuity and compact strategy spaces, we show that this condition implies the
existence of symmetric pure Cournot equilibria. We also state a set of stronger
sufficient conditions which in many applications may be easy to verify.
After establishing existence, one would also like to know when equilibrium is
unique. We show that asymmetric equilibria can be ruled out if the additional
weak condition that own market price reacts more to changes in own output than
in competitors’ outputs (condition B) is added. On the other hand, for competitor aggregation symmetric equilibrium is unique if each symmetric equilibrium is
“diagonally stable” as defined below, and if there is a symmetric equilibrium that
is unstable and “diagonally regular” then multiple symmetric equilibria exist. We
define these notions in developing an ordinal approach to stability and regularity of
symmetric equilibria. This approach does not need differential calculus, Jacobians
and determinants, and turns out to be very intuitive.
One of the fundamental conclusions of the oligopoly literature is that stability
of equilibrium is closely connected to “non-paradoxical” comparative statics results.
Our analysis for differentiated goods, which is based on ordinal monotone comparative statics methods, makes precise comparative statics predictions for extremal
(maximal and minimal) equilibria, which do not rely explicitly on stability conditions since extremal equilibria are generically stable. To deal with interior equilibria,
we invoke our results on stability and regularity of equilibria. The comparative statics of entry using these methods does not rely on treating the number of firms as a
continuous variable. Still, we show how each equilibrium, be it stable or not, can be
1
The instruments of lattice theory were introduced into economics by Topkis (1979), Milgrom
and Roberts (1990) and Vives (1990), and ordinal versions by Milgrom and Shannon (1994) and
Milgrom and Roberts (1994).
1
associated in a natural way with a new “neighboring” stable equilibrium, and that a
move to a new equilibrium resulting in counter-intuitive comparative statics means
that the new equilibrium is unstable.2
We concentrate exclusively on the comparative statics of exogenous entry. Our
main results are that if competitors’ quantities enter inverse demand aggregated
into one single number, then equilibrium prices do not increase as more firms enter
the industry, and that the aggregate of competitors’ outputs is strictly increasing.
We also show by means of an example that this result is not extendable to general
differentiated goods with aggregation into two or more numbers, i.e. equilibrium
prices may rise even if there are no increasing returns to scale and equilibrium is
stable.
As concerns quantities at equilibrium, total equilibrium output may rise or fall
even if prices are decreasing, but will rise under the same condition that we already
used to rule out asymmetric equilibria and an additional condition on the aggregator.
We also confirm the known results that individual output rises or falls depending
on whether goods are strategic complements or substitutes, while profits always
decrease with entry.
The rest of our paper continues as follows: Section 2 sets out the model, section
3 introduces the main condition, and the existence results are presented in section
4. Section 5 presents our comparative statics results and the discussion of stability.
Section 6 discusses the existing literature on existence and comparative statics of
Cournot equilibria, and section 7 concludes. All proofs are relegated to the appendix.
2
The Model
There are n firms with identical production capacities K and identical production
cost functions c : [0, K] → R+ . Denote firm i’s output quantity by xi , and by x−i the
vector of outputs of the other firms. Inverse demand of firm i (i = 1, .., n) is given by
a function p : [0, K]n → R+ , with pi = p (xi , x−i ),3 which is symmetric in the other
firms’ outputs: Let x̃−i be any permutation of x−i , then p (xi , x̃−i ) = p (xi , x−i ) for
all (xi , x−i ) ∈ [0, K]n . That is, firms are completely symmetric in that, apart from
identical production technologies, all demand functions have the same functional
form and all competitors’ goods enter each demand function symmetrically. Assume
that p is nonincreasing in xi and x−i , i.e. in particular goods are substitutes, and
strictly decreasing in xi where inverse demand is positive. We impose the following
regularity conditions:
Condition R (Regularity): 1. Production capacity K is limited: 0 < K < ∞; 4
2
Echenique (2001) contains related work for games with complementarities.
By p (xi , x−i ) we mean that xi is the first argument of p, i.e. that for all i own quantity xi
enters firm i’s inverse demand differently from other firms’ quantities xj , j 6= i.
4
Alternatively, as is often done, one may assume that inverse demand falls below marginal cost
(given any output of rivals) or even becomes zero for outputs larger than a certain limit. All these
3
2
2. production cost c (xi ) is continuous;5
3. inverse demand p (xi , x−i ) is continuous in (xi , x−i ).
Firm i’s profits are, for xi ∈ [0, K] and given x−i ∈ [0, K]n−1 ,
Π (xi , x−i ) = xi p (xi , x−i ) − c (xi ) , i = 1 . . . n.
(1)
We will now reformulate profits Π as a function of firm i’s market price pi and the
other firms’ outputs6 . This is done by introducing the residual demand function χ.
Lemma 1 There is a set X ⊂ R+ × [0, K]n−1 such that firm i’s residual demand is
given by a function χ : X → [0, K] of own market price pi and the other firms’ output
x−i that is continuous and nonincreasing in (pi , x−i ) ∈ X, and strictly decreasing
in pi . Also, there is a new constraint set π (x−i ) for firm i’s maximization problem
that is non-empty, closed, compact, and nonincreasing in x−i .
The very last fact follows from the assumption that goods are substitutes, and
is necessary for the conclusion below that equilibrium prices are non-increasing.
Firm i’s maximization problem can be expressed as
max Π̃ (pi , x−i ) = χ (pi , x−i ) pi − c (χ (pi , x−i )) ,
pi ∈π(x−i )
(2)
resulting in the price best reply7 P (x−i ).
If we consider profits Π̃ (pi , x−i ) ”on the diagonal” where all competitors produce
the same amount y ∈ [0, K], we write
Π̂ (pi , y) = Π̃ (pi , y, . . . , y) .
(3)
Some special cases are, in order of increasing specialization, what we will call “Competitor aggregation”, “industry aggregation”,8 and homogeneous goods. Under competitor aggregation there are functions p̂ : R2+ → R+ and f : [0, K]n−1 → R+ , where
f is strictly increasing, such that
p (xi , x−i ) = p̂ (xi , f (x−i )) ,
(4)
where the competitors’ quantities are aggregated into one number. One example is
additive aggregation with f (x−i ) = Yi = Σj6=i xj .
assumptions ensure that firms’ chosen outputs have a finite upper bound.
5
Lower semi-continuity of c already guarantees that each firm’s maximization problem has a
solution, but condition A below rules out upward jumps in costs anyway.
6
Note that the variable to be maximized over (output or price) is irrelevant as long as afterwards
each firm ’commits’ to a fixed production quantity or at least competitors believe that it is so.
7
We adopt this formulation to avoid confusion with the standard (quantity) best response or
reaction function xi = r (x−i ).
8
I would like to thank Karl Schlag for proposing these terms.
3
Under industry aggregation there exist functions p̄ : R+ → R+ and f : [0, K]n−1 →
R+ such that
p (xi , x−i ) = p̄ (xi + f (x−i )) ,
(5)
and if goods are homogeneous then f (x−i ) = Σj6=i xj . For industry aggregation it
is easy to see that χ (pi , x−i ) = D (pi ) − f (x−i ), where D = p̄−1 is the demand
function, and the profit maximization problem becomes
max
pi ∈[p̄(K+f (x−i )),p̄(f (x−i ))]
Π̃ (pi , x−i ) = (D (pi ) − f (x−i )) pi − c (D (pi ) − f (x−i )) .
In general it may not be possible to aggregate competitors’ quantities into one single
number as under competitor aggregation. In section 5 we state an example where
quantities are aggregated in two numbers, and in section 6.3 of Hoernig (2000) it is
shown that there are consumer’s preferences giving rise to a set of inverse demand
functions that do not allow for any aggregation of rivals’ quantities.
3
The Condition
In this section we will introduce several conditions on profits which all guarantee the
existence of equilibria. Even though the weakest one, condition A, seems somewhat
abstract at first sight, it nicely uncovers the economic intuition behind our and
previous existence results. Since condition A is in general not straightforward to
verify, we then state the two strictly stronger conditions AD and D, which are more
easily checked.
The main condition to be used for the proof of existence of equilibrium is of a
type that has recently been shown to be of central importance in any exercise of
comparative statics: In a lattice-theoretic context, Milgrom and Shannon (1994)
have shown that the set of maximizers of the parametric maximization problem
maxx∈S f (x, t), where S ⊂ R, is nondecreasing in (t, S) if f satisfies the weak single
crossing property (WSCP) in (x, t): for all x0 > x and t0 > t we have that
f (x0 , t) − f (x, t) ≥ (>) 0 ⇒ f (x0 , t0 ) − f (x, t0 ) ≥ (>) 0.
In the context of game theory this result can be applied to best reply maps, and we
do so after our change of variables from own quantity to own price described above.
Underlying our results is the following condition (It is a ”dual” single-crossing
condition because the inequality signs in the definition are reversed):
Condition A: Π̂ (pi , y) satisfies the dual weak single crossing property in (pi , y), i.e.
for all p0i > pi and y 0 > y we have that
Π̂ (p0i , y) − Π̂ (pi , y) ≤ (<) 0 ⇒ Π̂ (p0i , y 0 ) − Π̂ (pi , y 0 ) ≤ (<) 0.
(6)
Even though this condition seems to be extremely abstract, its interpretation is
4
very simple and economically intuitive: Condition A means that, starting from a
situation where all other firms produce identical quantities, if these other firms all
raise their outputs by the same amount, it will be advantageous for firm i to adjust
its output only in such a way that the resulting market price is not higher than before.
Doing so, own output may increase or decrease, depending on whether goods are
strategic substitutes or complements.
Note that condition A imposes the dual single crossing property only on the
”diagonal”, i.e. where competitors all produce the same quantity. This is sufficient
for the existence result since we are only interested in symmetric equilibria, while
we will have to state a condition covering the whole space of outputs to deal with
asymmetric equilibria. In addition, condition A is formulated for identical increases
in the outputs of all competitors. This is equivalent to formulating the corresponding
condition in terms of an increase in just one competitor’s quantity as long as inverse
demand is symmetric in competitors’ outputs, while it is more general if inverse
demands are not symmetric.9 Even though our condition (and the proof of existence
of symmetric pure equilibrium) applies to these more general cases, in this paper we
will concentrate on the symmetric case.
Condition A means that best reply price are non-increasing in the other firms’
outputs. In the proofs of theorem 2 (existence of equilibrium) and lemma 4 (monotonicity of the aggregator) below we only make use of an even weaker requirement
which follows from condition A:
Condition AW: Extremal best price replies ”on the diagonal” are continuous but
for downward jumps.
This means that downward jumps in own price are allowed but not upward
jumps. Clearly this follows when the price best replies are non-increasing. Still, we
need condition A for the comparative statics of prices, and also condition AW is less
intuitive.
We have stated what condition A means; still we should say what it rules out.
First, condition A and AW rule out strongly increasing returns to scale in production. The argument runs like this: Under increasing returns to scale, a reduction
in own output raises marginal cost, therefore creating incentives for further output
reduction. Therefore, if returns to scale are large, best reply output may be reduced so much that market price actually rises. In particular, note that condition
A and AW rule out the existence of avoidable fixed cost, i.e. fixed costs that are
not incurred if nothing is produced: Firm i may prefer to stop producing at all
(possibly creating an upward jump in own price), instead of lowering its own price,
if the other firms raise their outputs. Any other upward jump in production cost is
9
Condition A therefore even applies to cases where firms are symmetric when all competitors
produce identical outputs, i.e. for all (x, y) ∈ [0, K]2 , pi (x, y, .., y) = pj (x, y, .., y), while demand
functions may differ ”off-diagonal”. It also applies to the case where firms are symmetric, but are
not affected equally by every competitor. One example of this last case are firms with only two
direct neighbors in some kind of ”circular city” model.
5
similarly excluded. Second, for general differentiated goods, demand may be such
that condition A is violated even if marginal cost is constant. We show below that
in this case condition A relates the slopes of the demand function with respect to
own and others’ quantities.
Condition A applies no matter whether inverse demand and production costs
are differentiable or not, nor does it appeal to quasiconcavity of profits. Since it
is an ordinal condition, it is not surprising that there is no equivalent condition in
terms of derivatives even if demand or cost are differentiable. Using the method of
dissection discussed in Milgrom and Shannon (1994, p. 167), we can find a sufficient
differential condition that is slightly stronger than condition A.10 If Πii and Πij are
the second partial derivatives of the profit function of firm i with respect to outputs,
and pi and pj the partial derivatives of the inverse demand of firm i with respect to
xi and xj , condition A is implied by
Condition AD: For all i, xi ∈ [0, K], and x−i = (y, ..., y) ∈ [0, K]n−1 ,
Πii (xi , x−i ) −
pi (xi , x−i ) ij
Π (xi , x−i ) ≤ 0.
pj (xi , x−i )
(7)
It is easily seen why condition A follows from condition AD. Consider a change dxj
in firm j’s output, and a corresponding change dxi in firm i’s output such that firm
i’s market price p remains constant, leading to the marginal rate of substitution in
demand MRS = −dxi /dxj = pj /pi . The total change in marginal profits Πi is then
pj
d i
Π (xi (xj ) , x−i ) = − i Πii + Πij .
dxj
p
Condition AD means that this expression is non-negative which indicates that firm
i must increase its quantity (or not change it) to return to an optimum. This in turn
at most lowers its price, as demanded by condition A.
We can express condition AD in an equivalent form which makes the relation
to existing results much clearer. Consider a change dxj in firm j’s output, and
a corresponding change dxi in firm i’s output such that firm i’s market price p
remains constant, resulting in the marginal rate of substitution in demand MRS =
−dxi /dxj = pj /pi . Let εMRS be the elasticity of this marginal rate of substitution
with respect to own output xi ,
εMRS =
pij pi − pii pj
∂ (pj /pi ) xi
=
x
,
i
∂xi pj /pi
pi pj
then condition AD can be restated as
Πii −
10
pi ij
Π = (1 − εMRS ) pi − c00 ≤ 0.
pj
See Hoernig (2000), appendix 9.2.
6
This formulation yields two further interpretations: There are at most “weakly”
increasing returns to scale (where “weakly” depends on the demand function), or
the profit margin p − c0 , “corrected for substitution”, is falling in own output.11
Condition AD is more likely to be violated if MRS is decreasing in xi (εMRS < 0),
and more likely to hold if MRS is increasing in xi (εMRS > 0). It is easily verified
that in the cases of industry aggregation or homogeneous goods εMRS = 0, and
that condition AD then reduces to Amir and Lambson’s (2000) homogeneous goods
condition p0 − c00 ≤ 0. In fact, it can be shown that industry aggregation is the
most general case of differentiated goods where εMRS = 0.12 As for competitor
aggregation, it can be shown that εMRS only depends on the properties of p (x, f )
and not on the properties of the aggregator f (x−i ) since the derivative of the latter
cancels out.
Condition AD also implies a lower bound on the slope of quantity best replies ri (x−i ),
¯
Πij
pj
dxi ¯¯
∂
ri (x−i ) = − ii ≥ − i =
.
(8)
∂xj
Π
p
dxj ¯p=const
For homogeneous goods this reduces to the well-known condition that the slope
of best replies must be at least −1. Therefore we have obtained a very insightful
interpretation of this classical slope condition, coinciding with the one we gave in
the previous paragraph: The slope of best replies must not be smaller than the MRS
of demand, which again leads to the result that own price does not rise after a best
reply to an increase in someone else’s output.
Apart from condition AD there are various other conditions that imply condition
A and that may sometimes be easier to verify. Some we will discuss in the next
section, but the one most easily verified is the following:13
Condition D: Π̃ (pi , x−i ) has (weakly) decreasing differences in (pi , x−i ), i.e.
¡
¢
¡
¢
Π̃ p0i , x0−i − Π̃ pi , x0−i ≤ Π̃ (p0i , x−i ) − Π̃ (pi , x−i )
(9)
for all p0i ≥ pi ≥ 0 and x0−i ≥ x−i ∈ [0, K]n−1 .
This condition is stronger than condition A (Milgrom and Shannon 1994). If inverse demand and production costs are twice continuously differentiable, it is equivalent to
∂2
Π̃ (pi , x−i ) ≤ 0,
∂pi ∂xj
11
(10)
An equivalent interpretation, due to Amir and Lambson (2000) for homogenous goods, is that
”inverse demand or price decreases faster (...) at any given output level than does marginal cost
at all lower output levels.”
12
If εMRS = 0 the partial differential equations pji = hj (x−i ) pii hold, with solution pi (xi , x−i ) =
F (xi + H (x−i )) for some functions F and H with ∂H/∂xj = hj .
13
Here x0−i ≥ x−i means that x0j ≥ xj for all j 6= i.
7
for all j 6= i, or (see Hoernig 2000 for the proof),
µ
¶
¡
¢ εMRS
pi ij
ii
Π − jΠ
− pi + xpi − c0
≤ 0.
p
xi
(11)
The last term in (11) disappears if goods are industry aggregates (εMRS = 0), or at
interior best replies (pi + xpi − c0 = 0), leading to equivalence with condition AD in
these cases.
4
Existence of Equilibria
We will now state our main result on the existence of symmetric pure Cournot
equilibria. Multiple symmetric equilibria can be ranked according to equilibrium
quantities (or prices). If there is a symmetric equilibrium where quantities are
smaller (higher) than in any other symmetric equilibrium, this equilibrium is called
minimal (maximal).
Theorem 2 Assume that inverse demand is nonincreasing in all arguments (goods
are substitutes), and strictly decreasing in own output while inverse demand is positive. Under conditions R and A there exist symmetric pure Cournot equilibria.
The proof can be found in appendix 8.2. From a technical point of view, at
the heart of theorem 2 lies the fact that under condition A the price best reply
P (x−i ) has nonincreasing maximal and minimal selections, which allows for the
construction of a function on [0, K] into itself that is continuous but for upward
jumps and therefore has maximal and minimal fixed points (Milgrom and Roberts
1994).14 These result in maximal and minimal symmetric pure strategy Cournot
equilibria.
The equilibrium is unique if and only if the maximal and minimal equilibria are
identical, but this cannot be established without further assumptions. Under the
assumption that best replies are single-valued, which follows for example from the
strong assumption that profits are quasi-concave, in section 5 we show that there is
at most one symmetric equilibrium if all symmetric equilibria are stable, and that
there are several symmetric equilibria if one of them is unstable (see proposition 8).
These results are deferred to the section on entry because they are preceded by an
in-depth discussion of the stability issue using ordinal methods.
On the other hand, using stronger versions of condition A and a weak additional
condition B, one can exclude the existence of asymmetric equilibria. Let us state
14
In a previous version, we constructed a nondecreasing map from the space of prices into itself.
Tarski’s (1955) theorem can then be applied to show that maximal and minimal fixed points exist
(see Hoernig 2000 ).
8
two conditions related to condition A, both of which are strictly stronger and involve
the whole space of competitors’ outputs [0, K]n−1 : For all i = 1...n,
Condition AS: Π̃ (pi , x−i ) satisfies the dual strict single crossing property in
(pi , x−i ) for all x−i ∈ [0, K]n−1 : For all p0i > pi and x0−i > x−i ,
¡
¢
¡
¢
(12)
Π̃ (p0i , x−i ) − Π̃ (pi , x−i ) ≤ 0 ⇒ Π̃ p0i , x0−i − Π̃ pi , x0−i < 0.
Condition ASD: ∂ Π̃/∂pi exists and is strictly decreasing in xj for all j 6= i where
p (xi , x−i ) is positive and for all x−i ∈ [0, K]n−1 .
Condition AS means that a firm will not raise any best reply market price as a
reaction to an increase in competitors’ outputs (as opposed to just maximum and
minimum best reply prices under condition A). Condition ASD implies that a firm
will strictly decrease its market price as a reaction to an increase in competitors’
outputs and is stronger than conditions A and AS, and even stronger than weakly
or strictly decreasing differences of profits (see Edlin and Shannon 1998b).
Let x−ij be the vector of outputs of firms k 6= i, j. The additional conditions on
inverse demands are:
Condition BW (weak): For all j 6= i, all (xi , xj , x−ij ) ∈ [0, K]n , and all ε > 0,
p (xi + ε, xj , x−ij ) ≤ p (xi , xj + ε, x−ij ).
Condition BS (strict): For all j 6= i, all (xi , xj , x−ij ) ∈ [0, K]n , and all ε > 0,
p (xi + ε, xj , x−ij ) ≤ p (xi , xj + ε, x−ij ), where the inequality is strict when p(xi , xj +
ε, x−ij ) > 0.
If inverse demand is differentiable these conditions correspond to pi ≤ pj and
pi < pj (almost everywhere), respectively. Conditions BW and BS mean that each
firm’s changes in quantity influence its own market price more than the same changes
in other firms’ quantities, which is a very reasonable assumption as firms are symmetric. Note that the case of homogeneous goods, where pi = pj = p0 , falls under
condition BW. In fact, both these conditions follow from utility maximization of a
representative consumer: If inverse demands are derived from maximizing a (strictly)
concave utility function U, where at the optimum pi = ∂U/∂xi , then the condition
pi ≤ pj (pi < pj ) follows from the (strict) negative definiteness of the Hessian and
the symmetry of the demand functions.
With these conditions, we have the following proposition:
Proposition 3 Asymmetric equilibria do not exist if either conditions AS and BS
hold, or if conditions ASD and BW hold.
For homogeneous goods we must assume condition ASD (including the assumption that profits are differentiable) to rule out asymmetric equilibria, while for differentiated goods the weaker condition AS is enough. On the other hand, condition
AS must be accompanied with the slightly stricter condition BS.
9
5
Stability and Effects of Entry
A long-standing point of interest has been the question whether Cournot equilibrium approaches competitive equilibrium as more firms enter the market.15 It has
become common to call a Cournot equilibrium quasi-competitive if in equilibrium
total quantity is increasing or price is decreasing in the number of firms (independently of where price actually converges). It is easy to see that for differentiated
goods there is not necessarily a strict inverse relation between total quantity and
market prices: Outputs are not of the same kind, thus it is not really clear what
the economic meaning of ’total output’ is, while preference for variety may actually
lead to higher prices. Therefore it makes most sense to define quasi-competitiveness
through decreasing prices.
In our framework of horizontal differentiation, the entry of a new competitor
raises the number of goods (and welfare if consumers value variety), which in general may have surprising effects. As we will see in the following, under competitor
aggregation and condition A the conventional wisdom (equilibrium prices decrease
after entry) prevails, while this is no longer true in general.
Assume that there is a countable number of identical firms that may enter the
market16 . Let the aggregator f : [0, K]∞ → F ⊂ R be continuous, strictly increas∞
ing, and symmetric in its arguments: Let x̃−i be a¡ permutation of
¢ x−i ∈ [0, K] ,
then f (x̃−i ) = f (x−i ). For n ≥ 1 let g (x, n) = f (x)n−1 , 0, 0, ... be the value of
the aggregator if n − 1 firms produce x and the other firms nothing; this function
is strictly increasing in both arguments while x > 0 and n > 1, and continuous in
x. For all 1 ≤ n < ∞ let the maximum of the aggregator for a finite number of
producing firms be f¯n = g (K, n), and let f = f (0). Under competitor aggregation
inverse demand is given by
pi = p (xi , f (x−i )) ,
(13)
where x−i ∈ [0, K]∞ and p : [0, K] × F → R+ is continuous and non-increasing
in (xi , fi ), and strictly decreasing in its first argument while inverse demand is
positive. We can write residual demand given f as χ (p, f ), and price and quantity
best replies given f as P (f ) and R (f ). Then quantity best replies given x−i are
r (x−i ) = R (f (x−i )), and we have R (f ) = χ (P (f ) , f ). In particular, since χ is
continuous in (p, f ) and strictly decreasing in p > 0, and the extremal selections
of P are non-increasing in f (thus have no upward jumps), maximal and minimal
selections of R exist and are continuous but for upward jumps.
Imagine that each of n firms faces the value f of the aggregator of the other
firms’ output and all firms choose the same best reply x ∈ R (f ). The value of the
new aggregator is given by the following correspondence, which is fundamental for
15
The most recent contribution for homogeneous goods, using the traditional differentiable
methodology, is Dastidar (2000).
16
Since we are not interested in determining an endogenous free entry equilibrium, fixed cost of
entry are irrelevant for our purpose.
10
our study of comparative £statics
¤ and stability below: Define the family of correspondences ψ n : [f , ∞) → f , f¯n , n ≥ 1 by ψ n (f ) = g (R (f ) , n).17 We show in the
£
¤
proof of the following lemma that all fixed points of ψ n lie in f , f¯n , and that f ∗ is
a fixed point of ψ n if and only if there is a symmetric equilibrium with n firms and
∗
∗
∗
quantity
© ª y , and f = g¡(y¢ , n). Note that for n = 1 by definition ψ 1 is defined only
on f , with value ψ 1 f = f . Therefore the properties of symmetric equilibria
correspond to the properties of the fixed points of ψ n .
Under condition A, the existence of symmetric Cournot equilibria already follows
from theorem 2, therefore the following comparative statics conclusions are not
empty. The following lemma states the comparative statics of the aggregator, and
holds even under the weaker condition AW.
Lemma 4 Under competitor aggregation and condition A the maximum and minimum equilibrium values f(n) of the aggregator are non-decreasing in the number of
firms n, and strictly increasing if monopoly is strictly viable.
This very insightful lemma is the generalization of the known proposition for
homogeneous goods that if r0 ≥ −1 the sum of competitors’ outputs Yi = Σj6=i xi
increases as more firms enter, which underlies most comparative statics results.
Clearly, for homogeneous goods total output must then be non-decreasing (and price
non-increasing): From r0 ≥ −1 it follows that any increase in Yi is accompanied by
an equal or smaller decrease in xi , such that Q = xi + Yi does not decrease. This
need no longer be true with differentiated goods, as the additional assumptions of
proposition 11 below indicate.
Our main comparative statics result on prices follows almost trivially from this
lemma:18
Theorem 5 (Quasi-competitiveness) Under competitor aggregation the following
hold:
1. Under condition A maximum and minimum equilibrium prices are nonincreasing in the number of firms n.
2. Under condition ASD, and if p (xi , x−i ) is strictly decreasing in xj for all j 6= i
while pi > 0, then maximum and minimum equilibrium prices are strictly decreasing
in n as long as they are positive and monopoly is strictly viable.
Two remarks are in order: First, as noted above, condition ASD, the condition
that ∂ Π̃/∂pi exists and is strictly decreasing in xj for all j 6= i, is strictly stronger
than condition A or even strictly decreasing differences of Π̃ in (pi , x−i ) (see Edlin
and Shannon 1998b). Thus the conclusion that extremal equilibrium prices are
17
The graph of ψ n , n ≥ 2, is a continuous transformation of the graph of the best response R
and has essentially the same shape, i.e. is homeomorphic.
18
The counter-examples to Seade (1980) of De Meza (1985) do not apply here since they do not
satisfy our condition A, just as they did not satisfy Seade’s condition A.
11
strictly decreasing (instead of non-increasing) needs rather more restrictive assumptions.
Second, both statements of theorem 5 do not rely on the implicit function theorem (IFT), neither do they treat n as a continuous variable; Both of these are
common for deriving comparative-statics conclusions. For general types of aggregation treating n as a continuous variable is impossible, and this was one of the
motivations for using the ordinal approach.19 The second statement includes differentiability in its assumptions, which is necessary to obtain a statement about strict
monotonicity, but still its proof does not make use of the IFT.
Our arguments so far can by their nature only produce conclusions about maximal or minimal equilibria, but not about equilibria “in between”. Here it is useful to
make a distinction between the comparative statics of the equilibrium point, i.e. “the
direction in which the equilibrium point moves”, and comparative statics of equilibrium, i.e. where starting at a given previous equilibrium point after a change in
parameters the “out-of-equilibrium dynamics” will converge (if it does). Equilibria
can be unstable, and in general counter-intuitive comparative statics of equilibrium
points are associated to these unstable equilibria.20 For instance, in our case it could
happen that the unstable equilibrium point moves towards higher prices. Yet, as
Dierker and Dierker (1999) and Echenique (2001 ) stress in the context of games
with complementarity, out-of-equilibrium dynamics will not follow the (downward-)
shifted unstable equilibrium. Rather, using lattice-theoretic arguments, Echenique
shows that some general classes of dynamics converge to a higher stable equilibrium,
providing an ordinal version of the correspondence principle. We can apply their
ideas to Cournot oligopoly even though in general it is not a game with complementarities.
There are several problems to consider in the case of Cournot oligopoly. First
of all, best replies may be at least locally decreasing, which creates the possibility
of cycles. This latter phenomenon is made even more likely if best replies are not
single-valued. To concentrate on the effect of slope, for the study of non-extremal
equilibria we assume that R and therefore ψ n are single-valued. Then since both
are upper hemi-continuous correspondences by condition R, they become continuous
functions.
A second problem is that stability must be defined with respect to a certain class
of out-of-equilibrium dynamics. We will concentrate on the continuous best-reply
dynamics (CD) ẋ = k (r (x−i ) − x), k > 0,21 but will consider this dynamics only for
equal outputs of all firms, giving rise to the concept of diagonal stability. Considering
the whole space of outputs, instability can arise for two reasons. For homogeneous
19
See also Seade’s (1980) arguments and De Meza’s (1985) critique.
For the traditional approach to stability using differential methods, see Seade (1980), AlNowaihi and Levine (1985) and Furth (1986).
21
This is not a contradiction to our use of the ordinal approach since we do not assume any
expression to be differentiable, and technically it is unproblematic since r is a continuous function
of x−i if it is single-valued.
20
12
goods, as Seade (1980) has shown, it is precisely the diagonal of equal outputs that
constitutes the “unstable manifold” (along which outputs diverge) if r0 > 1/ (n − 1),
or, as Furth (1986) has shown, the unstable manifold involves some outputs rising
and others decreasing if r0 < −1 holds.22 We can show that in our case the results
are qualitatively the same even without taking recourse to differentiability. Since
our purpose in this section is to compare symmetric equilibria, and also for lack of
space, we will deal exhaustively with these questions in a sequel to this paper.
Consider the following continuous aggregator dynamics (CAD)
£
¤based on the
˙
¯
maps ψ n : Let f = k (ψ n (f ) − f ) = k (g (R (f ) , n) − f ) for f ∈ f , fn . This means
that if the aggregator f increases then all firms respond with an output higher than
that implied by f . In other words, for equal outputs this is just a rewriting of the
best-reply dynamics in quantities, and it is easy to see that for n ≥ 2 we have ẋ T 0
if and only if f˙ T 0 at f = g (x, n), i.e. the dynamics of x and f do have the same
properties:
ẋ T 0 ⇔ R (g (x, n)) T x ⇔ g (R (g (x, n)) , n) T g (x, n) ⇔ ψ n (f ) T f ⇔ f˙ T 0.
Since the ψ n have useful properties we will use the dynamics of f instead of the
dynamics of x. It is clear that f˙ > (<) 0 if and only if ψ n (f ) > (<) f , and that at
equilibrium points f˙ = 0.
Now we introduce our definition of local diagonal stability (We will drop the
word “locally” in the following if no confusion can arise).
Definition 6 A Cournot equilibrium point (x∗ )n is locally diagonally stable (dstable) if there exists δ > 0 such that for all ε ∈ (−δ, δ) \ {0} the CD starting
at (x∗ + ε)n converges to (x∗ )n .
Clearly d-stability is a necessary (but not sufficient) condition for local stability
in the usual sense: If an equilibrium is not d-stable, then it is unstable.
It is easy to see that an equilibrium is d-stable if and only if ψ n (f ) > f in a
neighborhood below and ψ n (f ) < f in a neighborhood above the equilibrium point,
i.e. if ψ n strictly cuts the diagonal from above: ẋ > 0 is equivalent to f˙ > 0 which in
turn is equivalent to ψ n (f ) > f since ψ n is continuous in f , etc. We say that a fixed
point f ∗ of ψ n is d-regular if ψ n at f ∗ strictly cuts the diagonal either from above or
from below. By the above definition a d-stable equilibrium is also d-regular, while
if the equilibrium is unstable then either ψ n cuts the diagonal from below, or the
equilibrium is not d-regular, i.e. ψ n touches the diagonal and may or may not cut it.
Again, d-regularity is necessary but not sufficient for regularity in the usual sense.
Sufficient conditions on profits for d-stability (instability) can easily be obtained
(let e = (1, .., 1) ∈ Rn ):
22
For this latter case also see Amir and Lambson (2000). It can be shown that a strict version
of our condition A, together with condition BW and the assumption that f is element-wise convex
(which both hold for homogeneous goods) rules out this type of instability.
13
i
i
Condition S (US): ∂Π
exists, and there exists ε > 0 such that ∂Π
(x + δe) is
∂xi
∂xi
strictly decreasing (increasing) in δ for all δ ∈ (−ε, ε).
Condition S means that, starting from outputs xj , j = 1..n, if all competitors
raise (lower) output by the same amount δ to xj + δ then the best reply of firm i
is strictly smaller (higher) than xi + δ, ruling out explosive paths where all outputs
diverge upwards or downwards. Condition US on the other hand precisely requires
that outputs diverge in both directions. These conditions imply directly that at
the equilibrium point ψ n strictly cuts the diagonal from above (below), leading to
d-stability or instability as explained above:
Proposition 7 If condition S (US) holds at a symmetric equilibrium x∗ and the
equilibrium price is positive, then it is d-stable (d-regular and unstable).
The assumption that profits Π are differentiable in own output is necessary
because without it we cannot conclude that the equilibrium is d-regular in the first
place (see also the differences between conditions A, AS and ASD).
Intuitively, if there are multiple symmetric equilibria, some of them should be
unstable, at least if best replies do not have jumps. That is intuition is correct is
shown by the following proposition:
Proposition 8 Let best replies be single-valued.
1. If all symmetric equilibria are d-stable then there is at most one symmetric
equilibrium.
2. Under condition A, if condition US holds at some symmetric equilibrium then
there exist multiple symmetric equilibria.
This proposition says nothing about asymmetric equilibria (for this see proposition 3), neither does it exclude multiple symmetric equilibria if best replies are
set-valued.
Let us now turn to the comparative statics of symmetric equilibrium points, be
they interior or not.
Proposition 9 Let ψ n be a continuous function for all n ≥ 2 and monopoly be
strictly viable. For some given n ≥ 1 let fn∗ be a fixed point of ψ n , i.e. an equilibrium
aggregator with n firms:
∗
∗
1. For m > n there is a smallest fixed point fm
of ψ m above fn∗ , with fn <
∗
∗
∗
corresponds to a stable (or non-regular) equilibrium with
fn+1 < fn+2 < .... Each fm
m firms.
−
2. If for m > n there is a highest fixed point fm
of ψ m below fn∗ , highest fixed
−
−
points fl− below fn∗ exist for all n < l < m. These are ordered, fn∗ > fn+1
> ... > fm
,
and for each k = n + 1..m the corresponding equilibrium with k firms is unstable (or
non-regular).
14
Note that both statements hold no matter whether the equilibrium at hand itself
is stable or not, or even regular. It is purely based on the observation that after
the transition from ψ n to ψ n+1 we have ψ n+1 (fn∗ ) > fn∗ , and on the assumption that
ψ n+1 is continuous (but for upward jumps).
Statement 1 gives a precise meaning to the notion of “equilibrium is increasing
in n”, which is not at all clear when the number of firms is treated as a discrete
variable. It also includes in a natural way the result about maximal and minimal
equilibria in lemma 4, which were proved under less restrictive assumptions. Starting
∗
at fn∗ , the CAD dynamics
to the fixed point fn+1
¢ ψ n+1 clearly converges
£ ∗ under
∗
∗
since ψ n+1 (f ) > f on fn , fn+1 by definition of fn+1 . Thus in this sense there is
necessarily a “convergence along the diagonal” to the “next-highest” equilibrium if
the number of firms increases.
The second statement is related to corollary 3 in Echenique (2001), which made a
similar statement for games with complementarities. It means that if some method
of relating equilibria for different n to each other (for example by treating n as a
continuous variable) leads to the counter-intuitive comparative statics result that
“equilibrium is decreasing” then it must have picked unstable (or non-regular) equilibria as the number of firms increased.
Even so this does not mean that after entry of a new firm the new equilibrium will
∗
be at fn+1
since in general after entry the vector of outputs will not be on the diagonal. Still, we can show that the out-of-equilibrium dynamics of the incumbent firms
starts with an upward trend: Consider an equilibrium with n firms and aggregator
fn∗ = g (x∗n , n). A firm n + 1 will enter if its best reaction R (g (x∗ , n + 1)) is positive,
which directly leads to an increase in the aggregators faced by the incumbent firms,
towards the next stable equilibrium, and away from any smaller unstable equilibrium. Furthermore, convergence to any unstable equilibrium will almost surely not
occur. The question of where this dynamics actually converges is not one of local
stability, and is complicated by the fact that overshooting or undershooting may
occur, on the contrary to games with complementarities. A final answer must take
into account the global properties of the out-of-equilibrium dynamics.
We will now show how our results are related to the differential approach. The
above conditions are ordinal versions of the usual differential conditions as in Seade
(1980), and we can even define some type of regularity without recourse to Jacobians
and determinants. If profits are twice continuously differentiable, condition S (US)
is equivalent to
Πii + (n − 1) Πij < (>) 0,
which is precisely Seade’s (1980) condition for stability on p. 23 (instability condition
B2 or (12)). To make the relation to ψ n clear in the differentiable case, assume that
f (.) is differentiable with component derivative fx , and R is differentiable in f .
Then from condition S it follows that ∂ψ n /∂f < 1, and from ∂g/∂x = (n − 1) fx we
15
obtain
∂ψ n
1
= (n − 1) fx R0 < 1 or r0 <
or Πii + (n − 1) Πij < 0,
∂f
n−1
with the corresponding inequalities for condition US. Furthermore, in the differential
case the equilibrium is not d-regular if and only if Πii + (n − 1) Πij = 0 or r0 =
1/ (n − 1).
Deliberately forcing the classical method based on the implicit function theorem
to its limits, we assume that g (x, n) is differentiable in n and treat n as a continuous
variable. This is done only for illustrative purposes, and is the only time in this paper
where we do this.
Lemma 10 Under competitor aggregation, let inverse demand and production cost
be twice continuously differentiable, and let g (x, n) be differentiable in n. The derivatives with respect to n of the aggregator and price at an interior equilibrium are
df(n)
x(n) fx Πii
=
dn
Πii + (n − 1) Πij
µ
¶
x(n) pj
dp(n)
pi ij
ii
Π − jΠ ,
=
dn
Πii + (n − 1) Πij
p
(14)
(15)
Let us first look at the comparative statics for the aggregator. Since f is increasing in x, and at an interior equilibrium necessarily Πii ≤ 0, the aggregator increases
(decreases) in n if Πii +(n − 1) Πij < 0 (> 0). These two conditions are precisely the
differential conditions for d-stability and instability, respectively, as shown above.
Furthermore, if condition AD holds then an interior equilibrium point moves
towards lower prices if and only if it is d-stable, and higher prices if and only if it is
unstable.
Theorem 5 and corollary 10 do not extend to the case of non-aggregative demands, as the following example shows23 . Note that the traditional comparative
statics analysis based on the implicit function theorem is not applicable here since
inverse demand cannot be written as a differentiable function of the number of firms.
Assume there are n firms with production capacity K ≥ 1/2 and zero production
cost, and inverse demands are given by p (xi , x−i ) = max {0, 1 − xi − Σj6=i (1 − e−xi xj )}.
Symmetric equilibrium outputs are given by the first-order condition (sufficient
second-order conditions are satisfied)
¡
¢
2
2 − 2x − n + (n − 1) 1 − x2 e−x = 0,
which for each value of n ≥ 1 has exactly one solution x(n) ≤ 1/2. Therefore for each
n there is exactly one symmetric equilibrium which at the same time is minimal,
23
In section 9.4.4 of Hoernig (2000) we give an intuitive explanation, using a fixed point map in
prices, of why without aggregation into one number we cannot prove that prices are non-increasing.
16
maximal and interior. On the diagonal xj = x̄ (j 6= i) condition AD is fulfilled for
any number of firms since
pi ij
Π − j Π = −xx̄ < 0,
p
ii
and the equilibrium is d-stable according to the above definition since
¡
¢
2
Πii + (n − 1) Πij = −2 − 2 (n − 1) x 2 − x2 e−x < 0.
Still, equilibrium prices fall until n = 3, and then rise:
<Figure 1>
One might ask: Up to which levels of aggregation does theorem 5 extend? The
following example shows that it is ”tight” in the sense that even if competitors’
outputs are aggregated into two numbers then condition A does not imply nonincreasing
equilibrium¡ prices. Consider
©
¢ª the inverse demand function p (xi , x−i ) =
2
2 3
max 0, 1 − xi − Σj6=i xi xj − xi xj /2 , and zero production cost. Condition A
holds at the unique symmetric equilibrium for all n ≤ 189, being stable for all
n, but equilibrium market prices are increasing for all n ≥ 6.
There are three other variables of interest whose equilibrium values vary with
the number of firms: Total output, individual outputs, and profits. In supermodular games, i.e. games with complementarities, (smallest and largest) equilibrium
strategies rise with the number of players, see Topkis (1998), theorem 4.2.3. In
Cournot oligopoly the comparative statics of individual quantities, total quantities
and profits each depend on a different condition.
Proposition 11 Under competitor aggregation the following holds in the comparison between a symmetric equilibrium with n firms and one with m > n firms:
1. If the equilibrium aggregator increases, f(m) > f(n) , then
a) Individual outputs x(n) do not increase (decrease) if xp (x, f ) fulfills the DWSCP
(WSCP) in (x, f ) for all j 6= i, i.e. if goods are strategic substitutes (complements);
b) Individual profits Π(n) do not increase, and decrease strictly if p (x, f ) is strictly
decreasing in f .
2. If the equilibrium price is positive and does not increase, then total output
Q(n) = nx(n) does not decrease (increases strictly) if f is quasi-convex and condition
BW (BS) holds.
Finally, industry profits, i.e. the sum of profits of all firms in the industry, may
be increasing or decreasing, as already described by Seade (1980) for homogeneous
goods.
Proposition 11 does not depend on whether the equilibria in question are stable or
even regular, thus is valid for all symmetric equilibria. Lemma 4 and theorem 5 show
17
that for maximal and minimal equilibria the assumptions of proposition 11 hold,
therefore their comparative statics are as expected. Proposition 9 indicates that one
should expect them to hold as well for interior equilibria, because the aggregator
tends to be increasing, and price decreasing, while we make no assumptions about
how the equilibrium with more firms is actually established.
Lastly, we give an illustration of what quasi-convexity of f means: Two units of
good 2 instead of one unit of good 2 and one unit of good 3 result in a (weakly)
lower price of good 1, i.e. are a closer substitute for good 1. This implies a taste
for variety, and is automatically fulfilled for homogeneous goods or if there are only
two firms in the market.
6
Related literature
In this section we will give an overview of the existing literature on existence of
Cournot equilibria and comparative statics of entry, and show how it relates to this
paper.
From the viewpoint of game theory, Cournot equilibria exist under the assumptions that profits are concave or quasi-concave and quantities bounded, using the
general result that games with quasi-concave payoffs and compact strategy spaces
possess pure equilibria (see Friedman 1991 or Dasgupta and Maskin 1986). These
conditions involve strong indirect assumptions about demand and production costs,
therefore there have been many attempts to identify those features in the Cournot
oligopoly model that guarantee the existence of equilibria under weaker assumptions.
Most work has concentrated on economically meaningful conditions on demand
or cost, or both. Interestingly, practically all proofs of existence are related with
either one or the other of Hahn’s (1962) pair of stability conditions,
p0 + xp00 ≤ 0 ⇔ Πij ≤ 0,
p0 − c00 ≤ 0 ⇔ Πii ≤ Πij ,
(16)
(17)
which are thus seen to impose two different kinds of strong regularity on the model.
The most intuitive way to characterize both strands of literature is to express all
conditions used in terms of slopes of (quantity) reaction functions r (x−i ): Assuming
that profits are twice differentiable, we obtain r0 (x−i ) = −Πij /Πii . The second
strand of literature effectively assumes that this slope is bounded below by −1,
while strategic substitutes (complements) imply that r0 ≤ (≥) 0. Our condition AD
in general implies r0 ≥ −pj /pi , which is equal to −1 under homogeneous goods, and
larger than −1 under condition BS.
Condition (16) is on demand only, and means that marginal revenue does not
increase if competitors raise their outputs. This implies that goods are strategic
substitutes in the terminology of Bulow et al. (1985), i.e. that best quantity replies
are non-increasing. More generally, goods being strategic substitutes (complements)
18
follows from profits Π (xi , x−i ) having weakly decreasing (increasing) differences in
outputs (xi , xj ) for j 6= i, or Πij ≤ (≥) 0 under differentiability.
For homogeneous goods, Novshek (1985) shows that (possibly non-symmetric)
Cournot equilibria exist if goods are strategic substitutes. For general aggregative
games, i.e. ”competitor aggregation”, Corchón (1994 , 1996) effectively imposes both
of Hahn’s conditions (in my notation Πii < Πij < 0), and proves existence through
the concavity of payoffs, while Kukushkin (1994 ) only assumes strategic substitutes.
For strategic complements Vives (1990), shows existence of pure Cournot equilibria
for differentiated goods (symmetric if firms are symmetric) in the general context of
supermodular games.
The second strand, related to condition (17), initially imposed assumptions on
costs only, and proves the existence of symmetric equilibria with homogeneous goods.
McManus (1962, 1964 ) and Roberts and Sonnenschein (1976) show that symmetric
pure Cournot equilibria exist assuming that production costs are convex or linear,
while Szidarovsky and Yakowitz (1977 ) additionally assume that inverse demand
is concave. Kukushkin (1993), assuming convex costs, shows existence of pure symmetric equilibria if outputs are discrete variables.24 Amir and Lambson (2000), still
for homogeneous goods, directly assume p0 − c00 < 0, allowing for limited increasing
returns to scale in production, and prove existence of pure symmetric Cournot equilibria. Their work is important in several respects: It shows that the assumption of
convex costs can be relaxed, and that the relevant condition p0 − c00 < 0 is latticetheoretic in nature. As one can see from condition AD, our condition A is a direct
generalization to differentiated goods of this condition.2526
While condition A generalizes the second strand of the literature, its relation
with the first strand is not straightforward. For homogeneous goods the assumption
of strategic complements implies condition A if profits are (at least locally) concave.
Since profits are locally concave at interior best replies, this result captures the fact
that reaction functions certainly have slope larger than −1 if they are nondecreasing.
For differentiated goods this relationship is not clear.
Most of the above authors have only covered the case of homogeneous goods.
Kukushkin (1994) and Corchón (1994, 1996), assuming strategic substitutes, deal
with additive aggregation, i.e. where the sum of competitors’ outputs is relevant;
Vives (1990) allows for general non-homogeneous goods and strategic complements,
24
Mixed equilibria always exist if outputs are discrete.
Amir (1996b) obtains new results in both strands of the literature, imposing various logconcavity or -convexity conditions on demand and cost, and using the ordinal approach of Milgrom
and Shannon (1994).
26
Spence (1976) presents a class of demand functions with a special functional structure where
Cournot equilibria can be found maximizing a certain ’wrong’ surplus function, later called a
’potential’. Here the question of existence of Nash equilibria reduces to the question of existence of
maxima of this function. Slade (1994) finds a necessary and sufficient condition for the existence of
a potential, e.g. that for homogeneous goods demand must be linear. The literature on ’potential
games’ is further developed in Monderer and Shapley (1996) and Kukushkin (1999); interestingly,
the latter also uses an ordinal approach.
25
19
but if goods are strategic substitutes his approach works only if there are no more
than two firms. Our work is the first to address the question of existence of equilibria with differentiated goods in a general context that does not make use of the
assumption of strategic substitutes or complements. Rather, it is based on the second strand of literature and can be understood as a generalization of Amir and
Lambson’s (2000) work to differentiated goods.
Comparative statics on demand or cost variables for Cournot oligopoly have been
analyzed by many authors, among them Frank (1965), Dixit (1986), Corchón (1994,
1996), while comparative statics with respect to entry have been discussed by Frank
(1965), Ruffin (1971), Seade (1980b), Szidarovsky and Yakowitz (1982), Corchón
(1994, 1996), and recently by Amir and Lambson (2000). The case of differentiated
goods has been treated by Dixit (1986) for two firms, and by Corchón (1994, 1996) for
additive aggregation, but they as most other authors have imposed the assumption of
strategic substitutes from the outset, which is irrelevant for most comparative-statics
conclusions, in particular for the question of quasi-competitiveness. In fact, Amir
and Lambson have shown for homogeneous goods that the only relevant condition for
decreasing equilibrium prices and increasing equilibrium total quantity after entry
is that there are no strong increasing returns to scale or Hahn’s condition (17). Our
conditions AD and A are generalizations to differentiated goods of this condition.27
De Meza (1985) and Villanova, Paradis and Viader (1999) exhibit examples
where for n ≥ 2 equilibrium prices are decreasing (therefore quasi-competitive) but
are higher than the monopoly price. This outcome is entirely due to the assumption
of strongly increasing returns to scale in production that can only be exploited by
monopolists. Our condition A rules this out (as did Seade’s (1980) stability condition
A), therefore theorem 5 holds also for the transition from monopoly to duopoly. De
Meza also raises the interesting point that stability or ”observability” of equilibrium
does not imply decreasing equilibrium prices since the monopolist’s output choice is
by definition observable even if it lies on an “unstable” part of the reaction function.
We agree with his point of view, but again theorem 5 is not subject to this critique
since it does not depend on stability.
7
Conclusions
Using an ordinal approach, we established the existence of pure symmetric Cournot
equilibria for homogeneous or differentiated goods under a simple condition that
generalizes the condition of weakly increasing returns used in the literature for the
case of homogeneous goods. We were able to rule out asymmetric equilibria using a
27
An issue that is related but different from quasi-competitiveness is the issue of convergence of
equilibrium to the ’competitive price’. Ruffin (1971), following McManus (1964) and Frank (1965),
shows that the Cournot equilibrium price converges to the competitive price (defined as minimum
average cost) if there are diseconomies of scale, and do not converge if there are increasing returns
to scale.
20
weak additional condition, and after discussing stability and regularity of equilibria
from an ordinal point of view, establish a strict relation between a weak notion of
stability and multiplicity of symmetric equilibria under competitor aggregation.
Under the main condition (maximal and minimal) equilibrium prices are decreasing in the number of firms if competitors’ quantities enter inverse demand as
an aggregate, but may be increasing if competitors’ quantities are aggregated into
more than one number. For all equilibria, stable or unstable, there is sequence of
neighboring stable equilibria involving a higher value of the aggregator and lower
market prices. Total quantity increases with the number of firms under some additional conditions, while individual quantities increase (decrease) if goods are strategic complements (substitutes). Individual firm profits decrease after entry. Any
decrease in the aggregator or increase in price must then be caused by the choice of
an unstable equilibrium point, to which out-of-equilibrium dynamics (almost surely)
does not converge.
One topic for further research is to extend our methods (and maybe some results)
to cases where product differentiation is not symmetric, as e.g. in Hotelling models,
or to models with exogenous or endogenous quality differences. We have indicated
some possibilities for generalization in these directions. Second, similar results will
be obtained by applying corresponding conditions to models of differentiated goods
price competition.
21
Acknowledgements:
Thanks go to James Dow and Karl Schlag for their help and invaluable comments,
and to Dan Kovenock and two anonymous referees for their help in making this a
better paper. All remaining errors are mine.
22
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24
8
Appendix
8.1
Proof of Lemma 1
Proof. Let minimum and maximum prices be pK = p (K, .., K) and p0 = (0, .., 0),
and the interval of possible prices with outputs by the other firms given
π (x−i ) = [p (K, x−i ) , p (0, x−i )] , x−i ∈ [0, K]n−1 .
(18)
The set π (x−i ) is the new constraint set for the maximization over pi , obviously nonempty, compact and convex, and is descending (nonincreasing) in x−i (see Milgrom
and Shannon 1994) since p (., x−i ) is nonincreasing in x−i .
The range of possible combinations between market price and the others’ outputs
is
ª
©
(19)
X = (pi , x−i ) ∈ [pK , p0 ] × [0, K]n−1 |pi ∈ π (x−i ) .
Let x̄ (x−i ) be the maximum output that firm i will produce given that the other
firms are already producing x−i , either because this output is equal to capacity, or
because inverse demand becomes zero:
x̄ (x−i ) = min {K, min {x ∈ [0, K] |p (x, x−i ) = 0}} ≥ 0.
(20)
Then since p is strictly decreasing and continuous on xi ∈ [0, x̄ (x−i )], we can express
firm i’s output as a function χ : X → [0, K] that is continuous and nonincreasing
in (pi , x−i ) ∈ X, and strictly decreasing in pi with image [0, x̄ (x−i )] for each x−i ∈
[0, K]n−1 .28 .
8.2
Proof of Theorem 2
Proof. First we show that price best replies are well-defined given the regularity
conditions. Given any vector of outputs x−i ∈ [0, K]n−1 of the other firms, firm i’s
maximization problem is
max Π̃ (pi , x−i ) = χ (pi , x−i ) pi − c (χ (pi , x−i )) ,
pi ∈π(x−i )
where χ is continuous in pi , c is continuous, and
π (x−i ) = [p (K, x−i ) , p (0, x−i )]
is a non-empty compact set. Then Π̃ is a continuous function of pi on the compact
set π (x−i ) and therefore attains its maximum. Thus the price best reply P (x−i )
exists, where P : [0, K]n−1 → [pK , p0 ] is a correspondence that is symmetric in x−i
28
This is an application of a continuous version of the implicit function theorem.
25
since p (xi , x−i ) is symmetric in x−i . Now restrict P to the ’diagonal’ x−i = (y, ..., y),
y ∈ [0, K], and define
P̄ (y) = P (y, .., y) , y ∈ [0, K] .
Secondly, maximal and minimal price best replies in P̄ (y) are nonincreasing in
y: The constraint set π (x−i ) is decreasing (Milgrom and Shannon 1994), since both
p (K, x−i ) and p (0, x−i ) are nonincreasing in x−i . This follows from the assumptions
that goods are substitutes and that p (xi , x−i ) is continuous in x−i . Invoking this
fact and condition A, by Milgrom and Shannon’s (1994) monotonicity theorem the
set of maximizers P̄ (y) is decreasing in y. This implies in particular that maximum
and minimum selections of P̄ exist and are nonincreasing in y. Let P̃ : [0, K] →
[pK , p0 ] be a maximum or minimum selection, then P̃ is a nonincreasing function,
and therefore continuous but for downward jumps (we only need the latter).
Finally, we construct fixed points of a suitable map. Given identical outputs
y ∈ [0, K] for all competitors, quantity
best replies
r̃ : [0, K] → [0, K] ”on the
³
´
diagonal” are given by r̃ (y) = χ P̃ (y) , y, ..., y . Then r̃ is continuous but for
upward jumps, since χ is continuous in all arguments, and decreasing in its first,
while P (x−i ) is continuous but for downward jumps. Therefore r̃ has maximal and
minimal fixed points (Milgrom and Roberts 1994, cor. 1), which are equilibrium
outputs.
8.3
Proof of Proposition 3
Proof. Consider an asymmetric equilibrium x = (x1 , . . . , xn ), where w.l.o.g. x1 =
y + ε > x2 = y. Then x0 =(x2 , x1 , x3 , . . . , xn ) is also an equilibrium. For conciseness
we now suppress the arguments (x3 , . . . , xn ). Equilibrium prices for firm 1 are p =
p (y + ε, y) and p0 (y, y + ε), with p0 > p by condition BS or p0 ≥ p by condition BW.
First impose condition BS, leading to p0 > p. Under condition AS, i.e. under the
dual strict single-crossing property of firm i’s profit Π̃ in (pi , y) for all i = 1..n, all
selections of best price replies are nonincreasing, i.e. p0 ≤ p since y + ε > y, which
is a contradiction to p0 > p.
For the second statement impose condition BW, leading to p0 ≥ p. Since under
condition ASD for all i = 1..n the partial derivative ∂ Π̃/∂pi exists and is strictly
decreasing in xj (j 6= i), price best replies are strictly decreasing in xj (Theorem
2.8.5 in Topkis 1998). Therefore, since p is a best reply at y and p0 at y + ε > y, we
must have p0 < p, and again arrive at a contradiction.
8.4
Proof of Lemma 4
Proof. Consider the family of correspondences ψ n , n ≥ 1, as defined in the text,
and note that for all f ∈ [f , ∞)
f = g (0, n) ≤ ψ n (f ) = g (R (f ) , n) ≤ g (K, n) = f¯n
26
£
¤
since R (f ) ∈ [0, K] for all f , therefore all fixed points of ψ n lie in f , f¯n . As g is
continuous and strictly increasing in its first argument, the maximal and minimal
selections of ψ n , ψ max
and ψ min
n
n , are both continuous but for upward jumps. Clearly,
for each f the value of ψ n (f ) is non-decreasing in n, being constant only for f such
that R (f ) = 0. Also, since g is
£ strictly
¤ increasing in its second argument if x > 0,
¯
ψ n (f ) < ψ n+1 (f ) for all f ∈ f , fn such that ψ n (f ) >f , and ψ n (f ) = ψ n+1 (f )
if ψ n (f ) = f and n ≥ 2 (because then R (f ) = 0). For each n ≥ 1, by theorem 1
has a minimal fixed point fnmin , and ψ max
has
in Milgrom and Roberts (1994) ψ min
n
n
max
a maximal fixed point fn , which are non-decreasing
¡ ¢in n. For n =¡1 ¢the unique
fixed point is f . If monopoly is strictly viable then R f > 0, and ψ n f >f for all
n ≥ 2. In this case for n ≥ 2 all fixed points are larger than f , where ψ n is strictly
increasing in n, therefore the maximum and minimum ones are strictly increasing
in n.
Last but no least, we show that f is the value of the aggregator at an equilibrium
point with n firms if and only if it is a fixed point of ψ n . Let y ∗ be quantity at a
symmetric equilibrium with n firms, f = g (y ∗ , n), then ψ n (f ) = g (R (f ) , n) =
g (y ∗ , n) = f because y ∗ is a best reply. For the
f ∗ be a fixed point
¡ ¢ converse, let
∗
∗
∗
of ψ n . For n = 1, f = f , and any y ∈ R f fulfills g (y , 1) =f . For n ≥ 2,
g (x, n) is strictly increasing in x, therefore there is a unique y ∈ [0, K] such that
f ∗ = g (y, n). Then g (y, n) = f ∗ = ψ n (f ∗ ) = g (R (f ∗ ) , n), i.e. y is a best reply to
f ∗ and therefore an equilibrium output. Therefore, the largest and smallest fixed
points of ψ n are the aggregators for the largest and smallest symmetric Cournot
equilibria.
8.5
Proof of Theorem5
Proof. Under condition A, by lemma 4 the aggregator f in maximal and minimal equilibria is non-decreasing in n, and minimal and maximal best reply prices
P (f ) (which are the new equilibrium prices) are non-increasing in f , thus extremal
equilibrium prices are non-increasing in n.
As for statement 2, the maximal or minimal equilibrium aggregators are strictly
increasing in n as long as equilibrium prices are positive and monopoly is strictly
viable. If ∂ Π̃/∂pi exists and is strictly decreasing in xj (and thus in f ), by theorem
2.8.5 of Topkis (1998) extremal price best replies P (f ) are strictly decreasing in f
while prices are positive. Thus extremal equilibrium prices are strictly decreasing
in n.
8.6
Proof of Proposition 7
Proof. Consider a symmetric equilibrium with quantity x∗ produced by each of
the n firms. Let fδ = g (x∗ + δ, n). Under condition S, by theorem 2.8.5 of Topkis
(1998) the best quantity reply R (fδ ) to x∗ + δ by all competitors is smaller (higher)
27
than x∗ + δ if δ > (<) 0. Thus
fδ = g (x∗ + δ, n) > (<) g (R (fδ ) , n) = ψ n (fδ ) ,
and the equilibrium is d-stable. On the other hand, if condition US holds then
d-regularity and instability follow along the same lines.
8.7
Proof of Proposition 8
Proof. Assume that all symmetric equilibria are d-stable, i.e. ψ n strictly cuts the
diagonal from above at each f ∗ = g (x∗ , n) where x∗ is an equilibrium output. Since
ψ n is continuous this can only be true if there is exactly one symmetric equilibrium.
By condition A maximal and minimal equilibria exist. Assume now that at some
symmetric equilibrium x∗ condition US holds, then ψ n strictly cuts the diagonal
from below at f ∗ = g (x∗ , n). Since at the minimal equilibrium x∗min we know that
ψ n (f ) ≥ f from the left, and at the maximal equilibrium x∗max , ψ n (f ) ≤ f from
the right, we have that x∗min < x∗ < x∗max , and there are at least three different
symmetric equilibria.
8.8
Proof of Proposition 9
Proof. Since monopoly is strictly viable, we have ψ m (fn∗ ) > £fn∗ for ¤any m > n.
Thus ψ 0m (f ) = max {fn∗ , ψ m (f )} is a continuous function on fn∗ , f¯m and fulfills
the assumptions of theorem 1 in Milgrom and Roberts (1994) and has a smallest
∗
fixed point fm
> fn∗ . This fixed point is the smallest fixed point of ψ m (f ) above fn∗ .
Since the same argument can be used on any l > m > n, there is a smallest fixed
∗
∗
point fl∗ of ψ l above fm
. Now since ψ l (f ) > ψ m (f ) > f on [fn∗ , fm
], fl∗ also is the
∗
smallest fixed point of ψ l above fn . Therefore the smallest fixed points above fn∗
∗
exist and are ordered, fl∗ > fm
> fn∗ . The corresponding statement cannot be made
about “highest fixed points below fn∗ ”, because they may not exist at all.29 Still, if
−
−
−
−
for m > n a highest fixed point fm
below fn∗ exists then fm
= ψ m (fm
) > ψ l (fm
)
−
for all l between n and m. By continuity ψ l must have a fixed point fl somewhere
−
in (fm
, fn∗ ). By the same logic as above these fixed points are ordered.
∗
(highest lower fixed
Consider now for m > n a smallest higher fixed point fm
−
∗
∗
point fm ). Since ψ m is a continuous function and ψ m (fn ) > fn , it cuts the diagonal
∗
−
at fm
from the left (at fm
from the right) and from above, therefore the equilibrium
∗
−
at fm (fm ) is either d-stable (unstable) or non-regular (We cannot say anything
about how ψ m cuts from the other side).
29
Therefore the application of the “local converse” of theorem 1 in Milgrom and Roberts (1994),
as they propose on p. 455, is somewhat problematic.
28
8.9
Proof of Lemma 10
Proof. Denote the partial derivatives of inverse demand p (x, f ) with respect to x
and f as p1 < 0 and p2 ≤ 0, of f (x−i ) at a symmetric equilibrium with respect to
xj (j 6= i) as fx > 0 (therefore pi = p1 and pj = p2 fx ), and of g (x, n) with respect
to x and n as gx = (n − 1) fx > 0 and gn = xfx > 0, respectively. The first-order
necessary condition for an interior best reply in terms of quantity, and the second
derivatives of profits, are
Πi =
Πii =
Πij =
∂
Π (xi , x−i ) = p (xi , f (x−i )) + xi p1
∂xi
2
∂
Π (xi , x−i ) = 2p1 + xi p11 − c00 ,
∂x2i
∂2
Π (xi , x−i )
∂xi ∂xj
(xi , f (x−i )) − c0 (xi ) = 0,
= (p2 + xi p12 ) fx .
The partial derivatives of x (f, n) are xf = 1/gx and xn = −gn /gx , and the first
order condition can be expressed in terms of f(n) as
¡ ¡
¢
¢ ¡ ¡
¢
¢¢
¢
¡
¢
¡ ¡
Πi = p x f(n) , n , f(n) + x f(n) , n p1 x f(n) , n , f(n) − c0 x f(n) , n = 0,
leading to
df(n)
x f Πii
Πii x
¡ n
¢ = ii (n) x
= − ii
.
dn
Π + (n − 1) Πij
Π xf + p2 + x(n) p12
In terms of quantities,
¡
¡
¢¢
¡
¢¢
¡
¡
¢
Πi = p x(n) , g x(n) , n + x(n) p1 x(n) , g x(n) , n − c0 x(n) = 0,
then we obtain
x(n) Πij
dx(n)
= − ii
.
dn
Π + (n − 1) Πij
The total derivative of equilibrium prices is
¢
x(n) Πij
x(n) fx Πii
dp(n)
d ¡
+
p
=
p x(n) , f(n) = −p1 ii
2 ii
dn
dn
Π + (n − 1) Πij
Π + (n − 1) Πij
µ
¶
x(n) p2 fx
p1 ij
ii
Π
=
−
Π .
Πii + (n − 1) Πij
p2 fx
8.10
Proof of Proposition 11
Proof. Since the equilibrium aggregator f(n) is increasing in n, thus by the DWSCP
(WSCP) of xp (x, f ) in (x, f ) equilibrium output x(n) is non-increasing (non-decreasing)
29
in n. Since goods are substitutes, profits Π (x, f ) are non-increasing in f . Since f(n)
is increasing in n,
¢
¡
¢
¡
¢
¡
Π x(n) , f(n) ≥ Π x(n+1) , f(n) ≥ Π x(n+1) , f(n+1) ,
where x(n) and x(n+1) are the corresponding equilibrium outputs, and the first inequality expresses the fact that x(n) maximizes profits given f(n) . If p (x, f ) is strictly
decreasing in f then the second inequality is strict.
As for the sum of outputs, assume that Q(n+1) = (n + 1) x(n+1) < Q(n) = nx(n) ,
n
then x(n+1) < n+1
x(n) . Denote by (x)m the m-dimensional vector with all elements
equal to x ∈ R. Then we have
¡
£¡
¢ ¤¢
p(n+1) = p x(n+1) , f x(n+1) n
¶ ¸¶
·µ
µ
n
n
> p
x(n) , f
x(n)
n+1
n+1
n
¶
¸¶
·µ
µ
n
n−1
,
x(n)
x(n)
≥ p x(n) , f
n+1
n−1 n + 1
h¡
i´
³
¢
≥ p x(n) , f x(n) n−1 , 0 = p(n)
The first (strict) inequality follows from the fact that p (x, f ) is strictly decreasing
in x while positive, non-decreasing in f , and f (x) increasing in x; the second one
from a transference of output x(n) / (n + 1) from firm n + 1 to firm 1 under condition
BW, and the last one from the quasi-convexity of f , as we show below. Thus we
have concluded that if Q(n+1) < Q(n) then equilibrium price strictly increases. This
is ruled out by assumption, and we have arrived at a contradiction. Under condition
BS the second inequality is strict, and we can relax our counterfactual assumption
to Q(n+1) ≤ Q(n) while arriving at the same contradiction.
A function f : Rn → R is quasi-convex if f (Σni=1 λi ai ) ≤ maxi=1..n f (ai ) for
λi ≥ 0 for all i and Σni=1 λi = 1. Let ei ∈ Rn , i = 1..n, be vectors of one’s and
equal to zero at element i, e.g. e1 = (0, 1,h.., 1), and consider
ai = x(n) ei , i = 1..n.
i
¢
¡
Because of symmetry, f (ai ) = f (an ) = f x(n) n−1 , 0 for all i = 1..n. Let λ1 =
.. = λn−1 =
f
1
n+1
and λn =
(Σni=1 λi ai )
2
,
n+1
then
¶
µµ
¶
1
1
2
= f
(n − 2)
x(n) , (n − 1)
+
x(n)
n + 1 n + 1 n−1
n+1
µµ
¶
¶
n
n−1
= f
x(n)
x(n)
,
n+1
n−1 n + 1
h¡
i
¢
≤ f (an ) = f x(n) n−1 , 0 .
This proves the second statement.
30
Figure Legends
Figure 1: Rising equilibrium prices without aggregation.
31
Figure 1
32